Repr\'esentations d\'eterminantales effectives des polyn\^omes univari\'es par les matrices fl\`eches

TL;DR

This paper proves that any univariate polynomial with real coefficients can be represented by an effective determinantal form using arrow matrices, with the signature linked to its real roots and their multiplicities.

Contribution

It establishes the existence of effective determinantal representations for univariate polynomials and characterizes their signatures based on real roots and multiplicities.

Findings

Effective determinantal representations exist for all univariate polynomials with real coefficients.

The signature of the representation relates to the number of real roots and their multiplicities.

Arrow matrices are used to construct these representations.

Abstract

We first show the existence of an effective determinantal representation for any univariate polynomial with real coefficients. Then, we more precisely establish that any univariate polynomial with real coefficients has an effective determinantal representation with signature (r+s,s) if and only if it has at least r real roots with multiplicity. The effective determinantal representations we construct used arrow matrices.